1. the values increase without bound (towards infinity)
2. the values collapse (to zero)
3. the values change, but do not seem be (1) or (2)Julia sets are strictly defined as class (3) points, which just sort of drift around to other class (3) points and don't really go towards zero or infinity. Some Julia sets just consist of a bunch of disconnected points - any z you pick that is part of the set will move around to a different, but totally disconnected point. These are "dust" sets:
Other Julia sets are sort of a wiggly line that outlines one or more areas (the "inside") - these areas are all connected. These are called "solid":
There are also some which are borderline between the two - the Julia set is a wiggly line, all connected, but it doesn't outline anything. These sets are "dendritic" types:
Mandelbrot was looking for some sort of clue as to which c numbers made disconnected sets, and which made connected sets. It turns out that the test is easy. You just start with a z of 0+0i, the origin of the complex plane. This is called a "critical" point for this equation. If this point is class (1), the Julia set is of the dust type. If this point is class (2), the Julia set is of the solid type. And in the rare case where it's class (3), the Julia set is of the dendritic type.
The interesting thing is that if you plot all these critical points on the
screen, coloring based on whether you get a class (1), (2), or (3) Julia set,
you get a different fractal - sort of a "master" Julia set:
If you zoom in on a portion of a Julia set (any one), you will see the same detail, but repeated on a smaller and smaller scale. The detail doesn't change, just the size. Because the detail in the Mandelbrot set (there's only one) is an amalgamation of all Julia sets, the detail you see when you zoom in is based on the precise location you're zooming, so different areas zoomed in on will show different detail - logically enough, often similar in shape to Julia sets taken from that area. Mandelbrot detail is never the same twice, and there are some very exotic and bizarre shapes in the set. You can technically keep zooming forever, but these days one is limited to the precision of their computer and their patience.
